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Abstract
The index of a Lie algebra is an important algebraic invariant. In 2000, Vladimir Dergachev and Alexandre Kirillov defined seaweed subalgebras of $\mathfrak{gl}_n$ (or $\mathfrak{sl}_n$) and provided a formula for the index of a seaweed algebra using a certain graph, a so called meander.
 In a recent paper, Vincent Coll, Andrew Mayers, and Nick Mayers defined a new statistic for partitions, namely the index of a partition, which arises from seaweed Lie algebras of type A. At the end of their paper, they presented an interesting conjecture, which involves integer partitions into odd parts. Motivated by their work, in this paper, we exploit various index statistics and the index weight generating functions for partitions. In particular, we examine their conjecture by considering the generating function for partitions into odd parts. We will also reprove another result from their paper using generating functions.
Highlights
An integer partition λ is a weakly decreasing finite sequence of positive integers λ = (λ1, λ2, . . . , λr) [1]
Mayers defined a new statistic for partitions, namely the index of a partition, which arises from seaweed Lie algebras of type A
At the end of their paper, they presented a conjecture on the difference between the number of partitions of n into odd parts with an odd index and the number of partitions of n into odd parts with an even index
Summary
The parity of a certain statistic assigned to partitions into odd parts was considered in the study of seaweed Lie algebras. Mayers defined a new statistic for partitions, namely the index of a partition, which arises from seaweed Lie algebras of type A. At the end of their paper, they presented a conjecture on the difference between the number of partitions of n into odd parts with an odd index and the number of partitions of n into odd parts with an even index. We will provide some remarks and further conjectures in the last section
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